Problem 72 - Algebra A Combined Function

Question

Perform each indicated operation. $$ \frac{7}{8} \div \frac{1}{2} $$

Step-by-Step Solution

Answer

The result is \( 1 \frac{3}{4} \).

Step 1: Understand Division of Fractions

To divide by a fraction, you multiply by its reciprocal. So, the operation \( \frac{7}{8} \div \frac{1}{2} \) can be transformed into a multiplication by the reciprocal of \( \frac{1}{2} \), which is \( \frac{2}{1} \).

Step 2: Set Up the Multiplication

Rewrite the division problem as a multiplication: \( \frac{7}{8} \times \frac{2}{1} \).

Step 3: Multiply the Numerators

Multiply the numerators of both fractions: \( 7 \times 2 = 14 \).

Step 4: Multiply the Denominators

Multiply the denominators of both fractions: \( 8 \times 1 = 8 \).

Step 5: Simplify the Fraction

The result of the multiplication \( \frac{14}{8} \) can be simplified by dividing both numerator and denominator by their greatest common divisor, which is 2: \( \frac{14}{8} = \frac{7}{4} \).

Step 6: Convert to a Mixed Number, if Necessary

Convert \( \frac{7}{4} \) into a mixed number if required. Divide 7 by 4, which equals 1 with a remainder of 3, resulting in \( 1 \frac{3}{4} \).

Key Concepts

Reciprocal

Every fraction has a unique partner called its reciprocal. Understanding what a reciprocal is becomes very important when dealing with fraction division.
To find the reciprocal of a fraction, you simply swap its numerator and denominator. It's like flipping the fraction over.
For example, the reciprocal of \(\frac{1}{2}\) is \(\frac{2}{1}\) or just 2.

• Important to Note: A number multiplied by its reciprocal equals 1. With fractions, \(\frac{a}{b} \times \frac{b}{a} = 1\).
• This concept allows us to turn division problems into multiplication problems, making them easier to handle.

So, whenever you see division by a fraction, reach for its reciprocal, and switch to multiplication.

Multiplying Fractions

Once you're ready to multiply fractions, it's pretty straightforward. Multiplication of fractions is different from whole numbers but simple when broken down:

• Multiply the numerators: Take the numbers on top of the fractions and multiply them together.
• Multiply the denominators: Do the same with the numbers at the bottom.
• Put both results over each other to get a new fraction made from the products.

Another helpful tip is: Before multiplying, check if you can simplify the fractions, which can make your calculations even easier during multiplication.

Simplifying Fractions

After multiplying fractions, the result may sometimes not be in its simplest form. Simplifying fractions makes the numbers easier to understand.
When you simplify a fraction, you find a simpler version that has the same value. Here's how to do it:

• Find the greatest common divisor (GCD) of the numerator and the denominator.
• Divide both by this number.
• Your simplified fraction will have the smallest numbers possible while still representing the same amount.

For instance, with \(\frac{14}{8}\), both 14 and 8 can be divided by 2. So dividing the numerator and denominator by 2 gives you the simplified \(\frac{7}{4}\).

Mixed Numbers

Sometimes it's helpful to express an improper fraction as a mixed number, especially if the context calls for it, like in real-world scenarios:

• An improper fraction has a numerator larger than its denominator.
• A mixed number combines a whole number with a fractional part.

To convert, divide the numerator by the denominator to find how many whole numbers fit. The remainder then forms the new numerator for a fraction, with the old denominator.
For example, converting \(\frac{7}{4}\) means dividing 7 by 4, which gives you 1 whole, and the remainder of 3. So \(\frac{7}{4}\) can also be written as \(1 \frac{3}{4}\). This form is often easier to understand, especially when comparing sizes or amounts in practical situations.